4.5. Deligne-Mumford 叠

Last time we talked about separatedness. In particular, if is separated over , proper over , then is separated over .

The next topic is to work towards the first big theorem in stacks.

定义 If is Artin stack, then we say it is Deligne-Mumford (DM) if there exists scheme with .

In general, its not easy to check when a stack is DM. To check this, we recall the notion of formally etale/smooth. That is, we say is formally etale/smooth if for all diagramswith in , there exists unique arrow/exists arrow from to .

So, etale means exists unique arrow, smooth means exists arrow, and now we define formally unramified, means unique arrow.

定义 We say is formally unramified if for all diagramswith then .

定理 If is Artin stack over , then is DM iff is formally unramified.

We first state some corollaries, before we prove this.

推论 If is Artin stack over , then is algebraic space over iff for all , is trivial, i.e. algebraic spaces are Artin stacks with no stablizers.

证明. First, if is algebraic space, then is sheaf, i.e. fibered in sets, so no automorphisms.

Conversely, ifIf , then it is a scheme and is formally unramified. If , then it is an -torsor. That is, if we have two isomorphisms then are related by unique automorphism . By assumption, is the trivial group scheme, i.e. . So it is affine group scheme, and so is a scheme. Also, is a monomorphism, so it is formally unramified. Therefore is formally unramified and representable by schemes. Now by the big theorem above, we see there exists a etale cover.

Lastly, is a sheaf because the automorphisms are trivial and hence is algebraic space as desired.

注 Suppose is locally of finite presentation. Then the theorem says is DM iff for all and , the automorphism group is a finite group.

Why is this? We know is formally unramified iff is formally unramified for all . However, since is locally of finite presentation, is locally of finite presentation. Hence, we can check formally unramified for on geometric points, i.e. for all with we can check formally unramified.

If there is nothing to do, else (i.e. its non-empty) it is isomorphic to . But then is locally of finite presentation, so its formally unramified if and only if etale if and only if is finite if and only if is a group.

例 From the above remark, we see is not DM as is not finite. More generally, is not DM if is a positive dimension group scheme.

例 A long time ago we talked about moduli space of genus curves , where are given by with smooth proper and on geometric fibers it is genus curve.

When Deligne and Mumford defined DM stacks, they showed that for , is DM and defined a compactification . We will go through the idea of how to show is DM for .

We start with a curve of genus over a ACF . We want to be formally unramified (i.e. we want is finite group). So we want to show for all diagramswith , we have . Now, we consider the diagramwhere we get two arrows from to itself () and one arrow from to itself (), where both reduce to be over . Now by deformation theory, “” is a class in , where is tangent bundle. However, the degree of tangent bundles are given by . Hence we see for , the degree become negative, i.e. and hence , as desired.

What about ?

If , then is just over ACF , and over it is not. It is a Brauer-Severi variety. These are -torsors and so . We know , so is Artin, not DM. In particular, the dimension of is dimension of a point subtract dimension of , i.e. .

Next, we consider , the moduli space of genus curves with marked points. Let be a curve with and three marked points, then its isomorphic to with three additional points. Thus points.

In terms of deformation theory, we get lives in , where is twisted down by three points, thus the degree is .

Similarly, is Artin, but is genus curves with one marked point, which is just the moduli space of elliptic curves. In particular, , which is exactly the -invariant of elliptic curves. We get a map which sends elliptic curve to isomorphism class of (i.e. sends it to the -invariant).

We give a rough picture of what this looks like:


generically we have -stablizers since has automorphisms equal multiply by . However, for and , we have more automorphisms: and .

We don’t have time for the proof of the big theorme, but we go through the idea first.

If is a map of schemes, its smooth iff we can find Zariski cover withwhere and comes from the following: is locally free, we look locally on where is free. We choose basis which yields map to coming from .

For us, we have smooth , we would like to have something like “”. We look etale locally where is free to getFormally unramified will allow us to “slice” to get with etale arrow

Before we end, we talk about how to define .

We don’t realy know what to do, hence the first thing is to descent. Thus, consider the following diagramWe have and we get canonical isomorphism . Thus it satisfies the cocycle condition. Thus by descent of coherent sheaves, we get such that where is via the map .

Moreover, is locally free sheaf on because is.

In addition, we get . To show this, use descent:Thus we get .

We note that, for Artin stacks, this is usually not surjective. But it is surjective for DM stacks.

Last time we defined DM stacks to be Artin stack such that it has etale cover by a scheme.

定理 is DM iff is formally unramified.

Last time, for smooth we defined coherent locally free. We also mentioned the idea of the proof, which is to look at where is free, then we getThen using is formally unramified, we will “slice” until it becomes relative dim , i.e. etale.

Now we start the proof.

证明. We start with the easy direction. Assume is DM. Choose etale cover . Consider the following diagramThe goal is to show is formally unramified. Let be the projection, then we get the following diagramHowever, note is etale, hence is unramified, and hence is unramified as desired.

Conversely, suppose is formally unramified. Let be a ACF, . Choose with affine, such thatHowever, since is ACF, we get a section :Let be the residue field of image of in and let be the separable closure of .

Our goal is to show etale locally on and , we will find closed such that with . Then we are done as is etale.

Now we get the diagramwhich is the same as the following diagram

Now let . Last time we also constructed . This is usually not surjective, but we will show it is the case for DM stacks. To show it is surjective, by descent, it is enough to show surjective after applying . We see is formally unramified implies is formally unramified, and henceThis givesSince we have that map, it gives us that is surjective, and hence is surjective.

We havewhere we call the arrow as well. Locally, has image generates, so the same is true for .

Since is locally free, so we need to look etale locally where is free and there exists such that is a basis for .

Shrinking , we may assume . Let that generates , so we get . Then,and because is free for rank .

is smooth map, representable and relative dim in neighbourhood of , so it is etale in neighbourhood of . Shrink to assume is etale. Then . Then is etale implies the image is open. Let such that . Over , there exists such that .

In particular, we see Now, we import a fact, which is that, we can take an etale neighbourhood of such that there exists closed with and diagramthen there exists unique arrow with , i.e. we get the following diagramTherefore, we seeTherefore, by construction and henceNow is our desired map.

推论 If is a finite group, then DM.

证明. We just need groups are finite over . Over , -torsors are trivial, so our point isand group isSo . Then and hence we getIn other word, is stablizer of . Hence we getand so is finite as desired.

At the start of the course, we talked about general points determine conic. We did this through geometry on moduli space. In particular, moduli space of singular conics is exactly (if we drop singular, then this is not true!).

Even if we only interested in smooth curves, i.e. . To do intersection theory, we need a compactification . So, we need a notion of properness. To do this, we need quasi-coherent sheaves on .

定理 (Eisenbud-Harris). It is impossible to write down a general curve.

This uses intersection theory on . Note here we are asking for general curve. To see what this means, we consider the example of elliptic curve. For that, we know the short form for elliptic curve is given by . Thus it is the same as a dominant rational map , i.e. . They showed is of general type.

There is also another similar problem, where we look at , the moduli space of dim abelian varities. Then we know is of general type, is , and is unknown.